However, I do not understand this. I thought that momentum is a vector and that while the block is moving forward, the slab is moving backward, so I would need to have the following:

Mb*vb = Mb*vF + (Ms* -vF)

notice the negative vF for the final momentum term of the slab. I put a negative because the slab would be moving backwards with a negative velocity.

Why is my approach not correct? I got a different vF than 0.57.

Thanks

i think the first part is important -when you draw the FBD for the two bodies your confusion will get settled- remember when two surfaces in contact move relative to each other the frictional forces on the bodies also act equally but in opposite direction. if one is pulling behind the other will be pushing ahead. let us draw the part a, that is the forces.

I did the first part allright. For both of those objects, I will have a normal and gravitational force. the block will have a leftwards frictional force and the slab will have a rightwards frictional force.

The block is going to be going forward until the frictional force stops it.
The slab will thus have to go backwards to conserve momentum

I did the first part allright. For both of those objects, I will have a normal and gravitational force. the block will have a leftwards frictional force and the slab will have a rightwards frictional force.

The block is going to be going forward until the frictional force stops it.
The slab will thus have to go backwards to conserve momentum

The slab moves forward because there is nothing pushing the slab to the left. All the momentum is in the forward direction.

The slab moves forward because there is nothing pushing the slab to the left. All the momentum is in the forward direction.

How is that possible? I know that when I stand on a piece of cardboard and I move forward, the board will move backwards to conserve momentum [but I know that there is friction force here]. Are you saying that there is no friction between the block and slab that will cause the slab to move in reverse? [this is false because there is friction]

They say the speed of the slab and block is the same, so the magnitudes of their velocities are the same, but one should have a plus sign, but the other should have a negative one.

when I stand on a piece of cardboard and I move forward, the board will move backwards to conserve momentum

I assume you are thinking of standing still on the cardboard then walking forwards. That is not the same as set up here. The right analogy would be to get up speed on a (frictional) floor then jump onto the cardboard and try to stop on it. Which way will the cardboard move now?

First and foremost
The frictional force acting on the small block is in the backward direction (I assume the right direction to be the forward direction and the left direction accordingly)
The frictional force (which equals mu times N and is purely kinetic accelerates the slab
And the same frictional force which acts on the small block decelerates it to a certain speed vf
And the final velocity of the slab also becomes vf

No!!!
Why would the slab have a backward velocity?
When you run over a floor and gain speed
And then jump on a cardboard, the cardboard moves along with you
Why?
It's because of the points which are in contact (here, they're your legs)
Considering the cardboard offers friction
Your feet are moving forward w.r.t to the cardboard
However, consider the motion from your frame
When you move forward
Don't you momentarily see the cardboard moving behind you in the other direction? (However, this isn't the final speed of the cardboard, just the negative of the initial speed you possessed
And that too for a small instant
And now since w.r.t your feet, the cardboard tries to move backward, what will friction do?
It will oppose relative motion at the "point of contact" and push the cardboard forward!:)

Since there's no external force acting on the slab-block system (friction is an internal non conservative force)
Gravity is an external force here
But its component in the horizontal direction is m×g×cos 90=0
So using the principle of conservation of momentum
The answer you got is correct
Nothing wrong!!

I and the cardboard will move forward. But how do we know if this analogy is valid, especially with all those frictional forces we have drawn in the free body diagram?

What makes my approach wrong?

Friction acts to oppose relative motion (actual or potential) of the two surfaces in contact. Note that exact wording. So it acts on each of the two surfaces in opposition to its motion relative to the other surface.
The initial state is that the block is moving and the slab is not. Which two contacting surfaces are in relative motion? Which way does friction act on each to oppose that relative motion?